Estimation and Sufficiency Under the Mixed Effects Extended Growth Curve Model with Compound Symmetry Covariance Structure

Abstract

An extended growth curve model with fixed and random effects is considered. Under the assumption of multivariate normality, the maximum likelihood estimators of the fixed effects and the dispersion matrix are determined in a model with random nuisance parameters, both without any assumption on the covariance structure and under the assumption of compound symmetry. For this purpose, rules for differentiation of symmetric matrices are applied. Furthermore, when the experiments are designed in balanced complete blocks, particular symmetric matrices appear in the likelihood equations, allowing closed-form expressions for the estimators. It is also shown that the vector of sufficient statistics for the fixed effects extended growth curve model is also sufficient for the model with random nuisance parameters. The presented results are illustrated using a real data example.

1. Introduction

In 1964, Potthoff and Roy [1] introduced the growth curve model (GCM) as an extension of the multivariate analysis of variance model (MANOVA), suitable for modeling data with repeated measurements, where underlying trends or growth curves can be observed. Such examples can be found, for instance, in environmental or medical research, as well as in biostatistics. Assuming mutually independent, multivariate normally distributed observation vectors, the maximum likelihood estimators (MLEs) of the matrix of expectation parameters and the dispersion matrix of observations can be found, for example, in [2]. Note that when the dispersion matrix is proportional to the identity matrix, the multivariate normal distribution is symmetric and is called a spherical distribution; when the dispersion matrix is diagonal, the distribution is symmetric with respect to the corresponding eigenvectors. Furthermore, ref. [3] considered the GCM with a particular dispersion structure, namely compound symmetry, where all diagonal entries are the same and all off-diagonal entries are equal. Since such a matrix has only two distinct eigenvalues, the distribution represents the simplest generalization of the spherical normal distribution, because the observation variances remain equal, and the observations are equicorrelated rather than independent. Note also that the explicit formula for the MLE under such a structure, as well as the sufficient statistics for the estimation of the expectation and covariance structure, are given in [3].

A generalization of the GCM is the extended growth curve model (EGCM), also known as the sum of profiles model, introduced by [4]. The MLEs of the expectation matrices in an EGCM with two components, as well as the MLE of the dispersion matrix, can be found, for example, in [2]; while results for an EGCM with three components are given, for instance, in [2,5]. It is worth noting that explicit forms of the estimators can be determined if the nested subspace conditions are satisfied (cf. [2] for the estimators and examples). Explicit formulas for unbiased estimators of the expectation parameters and the dispersion matrix can be found in [6].

In all the aforementioned papers, fixed effects models have been considered. Although fixed effects models provide a straightforward way to account for systematic differences among groups, they are limited when the grouping factors represent only a sample of all possible conditions. In such cases, treating these factors as fixed restricts inference to the specific observed levels and may reduce the model generalizability. Mixed effects models address this issue by incorporating random effects that capture variation across a broader population of conditions.

The first aim of this paper is to show that, under multivariate normality, the vector of statistics sufficient for a fixed effects EGCM remains sufficient when the nuisance parameters are treated as random. The second objective is to determine the maximum likelihood estimators (MLEs) of the unknown parameters in a mixed effects EGCM. We first consider an EGCM in which both the covariance matrices of the random effects and the errors are positive definite matrices. Subsequently, we determine the estimators of the unknown parameters of a multivariate random block effects model with specific restrictions (which represents a special case of the EGCM), and under the assumption of compound symmetry of the error dispersion matrix. For the purpose of estimation, the rules for differentiation of symmetric matrices are applied. Furthermore, when the experiments are designed in balanced complete blocks, particular symmetric matrices appear in the likelihood equations, allowing the closed-form expressions for the MLEs. It is worth noting that parameter estimation in a mixed effects EGCM has been studied previously in, e.g., [3]; however, the assumptions of compound symmetry of the dispersion matrices of random effects and errors were imposed in that work.

This paper is organized as follows. In Section 2, we introduce the EGCM, we determine the MLEs of the parameters under the mixed effects EGCM, and we show that the vector of sufficient statistics for a fixed effects EGCM remains sufficient for a mixed effects EGCM. In Section 3, we consider a special case of the mixed effects EGCM, namely the model of block experiments with the same random block effects for every observation, subject to particular restrictions on the random nuisance parameters, and assuming compound symmetry of the dispersion matrix of errors. Similarly to the general mixed effects EGCM, we determine the MLEs of the unknown parameters and show that the vector of sufficient statistics for the fixed effects EGCM with a compound symmetry dispersion matrix remains sufficient for the mixed effect EGCM. We also show that, in the special case when the block design is complete, there exists the best linear unbiased estimator of the matrix of treatment effects, and the MLE of the compound symmetry dispersion matrix can be expressed in explicit form. In Section 4 we illustrate the results with real data examples. The paper ends up with short discussion.

2. Estimation Under Mixed EGCM

Consider the extended growth curve model (EGCM)

$$\mathbf{Y}\sim {\mathcal{N}}_{n,q}({\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1+{\mathbf{A}}_2{\mathbf{B}}_2{\mathbf{C}}_2,{\mathbf{I}}_n,\mathbf{\Sigma }),$$

(1)

where $\mathbf{Y}\in {\mathbb{R}}_{n\times q}$ is a matrix of normally distributed observations with independent rows, ${\mathbf{A}}_i\in {\mathbb{R}}_{n\times n_i}$ and ${\mathbf{C}}_i\in {\mathbb{R}}_{q_i\times q}$, $i=1,2$, are known matrices (the design and restrictions matrices, respectively), while ${\mathbf{B}}_i\in {\mathbb{R}}_{n_i\times q_i}$ are matrices of unknown parameters, $\mathbf{\Sigma }$ is an unknown symmetric positive definite matrix of order q, i.e., $\mathbf{\Sigma }\in {\mathbb{R}}_q^{>}$, such that $\mathbb{D}\left(\mathbf{Y}\right)=\mathbb{D}(vec\mathbf{Y})=\mathbf{\Sigma }\otimes {\mathbf{I}}_n$, with “vec” denoting the vectorization operator, which transforms a matrix into a vector by stacking its columns one below the other, and ⊗ denotes the Kronecker product; cf. [2]. It is worth mentioning that the matrices ${\mathbf{A}}_i$ usually contain information about the experimental designs, whereas the matrices ${\mathbf{C}}_i$ represent restrictions related to growth. An EGCM of the form (1) is called a fixed effect EGCM.

In model (1), the best linear unbiased estimators (BLUEs) of estimable functions of ${\mathbf{B}}_1$ and ${\mathbf{B}}_2$ usually do not exist, and thus the MLEs of ${\mathbf{B}}_1$ and ${\mathbf{B}}_2$, denoted by ${\tilde{\mathbf{B}}}_1$ and ${\tilde{\mathbf{B}}}_2$, under normality of the distribution of observations are usually considered. The log-likelihood function has the form

$$\begin{array}{ccc}\hfill L({\mathbf{B}}_1,{\mathbf{B}}_2,\mathbf{\Sigma };\mathbf{Y})& =& -\frac{nq}{2}ln\left(2\pi \right)-\frac{n}{2}ln|\mathbf{\Sigma }|\hfill \\ \hfill & & -\frac{1}{2}tr[(\mathbf{Y}-{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1-{\mathbf{A}}_2{\mathbf{B}}_2{\mathbf{C}}_2){\mathbf{\Sigma }}^{-1}\hfill \\ \hfill & & \phantom{-\frac{1}{2}tr[}\times {(\mathbf{Y}-{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1-{\mathbf{A}}_2{\mathbf{B}}_2{\mathbf{C}}_2)}^{\top }],\hfill \end{array}$$

(2)

where $|·|$ denotes the determinant, and, following [5], the normal equations are given by

$$\begin{array}{c}{\mathbf{A}}_1^{\top }(\mathbf{Y}-{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1-{\mathbf{A}}_2{\mathbf{B}}_2{\mathbf{C}}_2){\mathbf{\Sigma }}^{-1}{\mathbf{C}}_1^{\top }=\mathbf{0},\hfill \\ {\mathbf{A}}_2^{\top }(\mathbf{Y}-{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1-{\mathbf{A}}_2{\mathbf{B}}_2{\mathbf{C}}_2){\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2^{\top }=\mathbf{0},\hfill \\ n\mathbf{\Sigma }={(\mathbf{Y}-{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1-{\mathbf{A}}_2{\mathbf{B}}_2{\mathbf{C}}_2)}^{\top }·(\mathbf{Y}-{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1-{\mathbf{A}}_2{\mathbf{B}}_2{\mathbf{C}}_2).\hfill \end{array}$$

To obtain explicit estimators of all parameters, one of the common assumptions is $\mathcal{C}\left({\mathbf{C}}_1^{\top }\right)\subseteq \mathcal{C}\left({\mathbf{C}}_2^{\top }\right)$, where $\mathcal{C}(·)$ denotes a column space. Using the notation for orthogonal projectors ${\mathbf{P}}_A=\mathbf{A}{\left({\mathbf{A}}^{\top }\mathbf{A}\right)}^{-}{\mathbf{A}}^{\top }$, ${\mathbf{Q}}_A=\mathbf{I}-{\mathbf{P}}_A$, and ${\mathbf{P}}_{A;B}=\mathbf{A}{\left({\mathbf{A}}^{\top }\mathbf{B}\mathbf{A}\right)}^{-1}{\mathbf{A}}^{\top }\mathbf{B}$, ${\mathbf{Q}}_{A;B}=\mathbf{I}-{\mathbf{P}}_{A;B}$, the theorem below can be proved using the same techniques as in [5] (Theorem 4). The sketch of the proof is given in the Appendix A.

Theorem 1. 

In a fixed effects EGCM (1) with $\mathcal{C}\left({\mathbf{C}}_1^{\top }\right)\subseteq \mathcal{C}\left({\mathbf{C}}_2^{\top }\right)$ and an unknown positive definite covariance matrix Σ, the maximum likelihood estimators of the unknown parameters are given by

$$\begin{array}{ccc}{\mathbf{A}}_1{\widehat{\mathbf{B}}}_1{\mathbf{C}}_1\hfill & =& {\mathbf{P}}_{A_1;Q_{A_2}}\mathbf{Y}{\mathbf{P}}_{C_1^{\top };S_1^{-1}}^{\top }\hfill \\ {\mathbf{A}}_2{\widehat{\mathbf{B}}}_2{\mathbf{C}}_2\hfill & =& {\mathbf{P}}_{A_2}(\mathbf{Y}{\mathbf{P}}_{C_2^{\top };S_2^{-1}}^{\top }-{\mathbf{P}}_{A_1;Q_{A_2}}\mathbf{Y}{\mathbf{P}}_{C_1^{\top };S_1^{-1}}^{\top })\hfill \\ n\widehat{\mathbf{\Sigma }}\hfill & =& {\mathbf{S}}_2+{\mathbf{Q}}_{C_2^{\top };S_2^{-1}}{\mathbf{Y}}^{\top }{\mathbf{P}}_{A_2}\mathbf{Y}{\mathbf{Q}}_{C_2^{\top };S_2^{-1}}^{\top }\hfill \end{array}$$

where ${\mathbf{S}}_1={\mathbf{Y}}^{\top }{\mathbf{Q}}_{(A_1:A_2)}\mathbf{Y}$ and ${\mathbf{S}}_2={\mathbf{S}}_1+{\mathbf{Q}}_{C_1^{\top };S_1^{-1}}{\mathbf{Y}}^{\top }{\mathbf{P}}_{Q_{A_2}{A}_1}\mathbf{Y}{\mathbf{Q}}_{C_1^{\top };S_1^{-1}}^{\top }$.

It is known (see, e.g., [7]) that under the GMANOVA model—that is, the EGCM (1) with ${\mathbf{C}}_1={\mathbf{C}}_2={\mathbf{I}}_q$ – the vector

$$\mathbf{F}\left(\mathbf{Y}\right)=\left({({\mathbf{A}}_1:{\mathbf{A}}_2)}^{\top }\mathbf{Y},{\mathbf{Y}}^{\top }{\mathbf{Q}}_{(A_1:A_2)}\mathbf{Y}\right)$$

(3)

represents a sufficient statistic for the estimation of $({\mathbf{B}}_1,{\mathbf{B}}_2,\mathbf{\Sigma })$. It can be seen that this is also a sufficient statistic under model (1), as the log-likelihood function (2) can be expressed in terms of $({\mathbf{A}}_1^{\top }\mathbf{Y},{\mathbf{A}}_2^{\top }\mathbf{Y},{\mathbf{Y}}^{\top }\mathbf{Y})$.

Assume now that in the EGCM (1), the nuisance parameter ${\mathbf{B}}_2$ is random, with $\mathbb{D}\left({\mathbf{B}}_2\right)=\mathbb{D}(vec{\mathbf{B}}_2)=\mathbf{\Psi }\otimes {\mathbf{I}}_{n_2}$, where $\mathbf{\Psi }\in {\mathbb{R}}_{q_2}^{>}$ is an unknown dispersion matrix, and which is independent of the random error of the model. Then the EGCM with

$$\begin{array}{c}\mathbb{E}\left(\mathbf{Y}\right)={\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1,\hfill \\ \mathbb{D}\left(\mathbf{Y}\right)={\mathbf{C}}_2^{\top }\mathbf{\Psi }{\mathbf{C}}_2\otimes {\mathbf{A}}_2{\mathbf{A}}_2^{\top }+\mathbf{\Sigma }\otimes {\mathbf{I}}_n.\hfill \end{array}$$

(4)

is called a mixed effects EGCM.

Denote $\mathbf{y}=vec\mathbf{Y}$, ${\mathit{\beta }}_1=vec{\mathbf{B}}_1$, ${\mathit{\beta }}_2=vec{\mathbf{B}}_2$, and $\mathbf{v}=\mathbf{y}-({\mathbf{C}}_1^{\top }\otimes {\mathbf{A}}_1){\mathbf{\beta }}_1$. Let $\mathbb{D}\left(\mathbf{Y}\right)=\mathbf{\Omega }$. Moreover, let ${PTr}_u[·]$ denote the partial trace operator, which replaces each $u\times u$ block of a partitioned $mu\times mu$ matrix by its trace; cf. [8]. Then the MLEs of ${\mathbf{B}}_1$, $\mathbf{\Psi }$, and $\mathbf{\Sigma }$ can be determined as stated in the following theorem.

Theorem 2. 

In a mixed effects EGCM (4) with unknown positive definite covariance matrices Ψ and Σ, the maximum likelihood estimators of the unknown parameters can be obtained by solving

$$\begin{array}{c}({\mathbf{C}}_1\otimes {\mathbf{A}}_1^{\top }){\mathbf{\Omega }}^{-1}\mathbf{v}=\mathbf{0},\hfill \\ {PTr}_{n_2}\left[({\mathbf{C}}_2\otimes {\mathbf{A}}_2^{\top })({\mathbf{\Omega }}^{-1}-{\mathbf{\Omega }}^{-1}\mathbf{v}{\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1})({\mathbf{C}}_2^{\top }\otimes {\mathbf{A}}_2)\right]=\mathbf{0},\hfill \\ {PTr}_n\left[{\mathbf{\Omega }}^{-1}-{\mathbf{\Omega }}^{-1}\mathbf{v}{\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1}\right]=\mathbf{0}.\hfill \end{array}$$

Proof. 

The log-likelihood function can be written as

$$\begin{array}{ccc}\hfill L({\mathbf{B}}_1,\mathbf{\Omega };\mathbf{Y})& =& -\frac{nq}{2}ln\left(2\pi \right)-\frac{1}{2}ln|\mathbf{\Omega }|\hfill \\ \hfill & & -\frac{1}{2}{vec}^{\top }[\mathbf{Y}-{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1]{\mathbf{\Omega }}^{-1}vec[\mathbf{Y}-{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1]\hfill \\ \hfill & =& -\frac{nq}{2}ln\left(2\pi \right)-\frac{1}{2}ln|\mathbf{\Omega }|-\frac{1}{2}{\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1}\mathbf{v}.\hfill \end{array}$$

(5)

Note that $\mathbf{\Psi }$ and $\mathbf{\Sigma }$ (embedded in $\mathbf{\Omega }$) are symmetric. Therefore, differentiating the log-likelihood function with respect to symmetric matrices amounts to differentiating with respect to the lower triangles of these matrices, denoted respectively by ${\mathbf{\Psi }}^{▵}$ and ${\mathbf{\Sigma }}^{▵}$. By the chain rule given in [9], and following the differentiation formulas presented in [10], we obtain

$$\begin{array}{ccc}{\displaystyle \frac{\partial L}{\partial {\mathit{\beta }}_1}}\hfill & =& {\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1}({\mathbf{C}}_1^{\top }\otimes {\mathbf{A}}_1),\hfill \\ {\displaystyle \frac{\partial L}{\partial {\mathbf{\Psi }}^{▵}}}\hfill & =& [-{\displaystyle \frac{1}{2}}{vec}^{\top }{\mathbf{\Omega }}^{-1}+{\displaystyle \frac{1}{2}}({\mathbf{v}}^{\top }\otimes {\mathbf{v}}^{\top })({\mathbf{\Omega }}^{-1}\otimes {\mathbf{\Omega }}^{-1})]\hfill \\ \hfill & & \times ({\mathbf{C}}_2^{\top }\otimes {\mathbf{A}}_1\otimes {\mathbf{C}}_2^{\top }\otimes {\mathbf{A}}_2)({\mathbf{I}}_{q_2}\otimes {\mathbf{K}}_{n_2,q_2}\otimes {\mathbf{I}}_{n_2})({\mathbf{I}}_{q_2^2}\otimes vec{\mathbf{I}}_{n_2}){\mathbf{D}}_{q_2},\hfill \\ {\displaystyle \frac{\partial L}{\partial {\mathbf{\Sigma }}^{▵}}}\hfill & =& [-{\displaystyle \frac{1}{2}}{vec}^{\top }{\mathbf{\Omega }}^{-1}+{\displaystyle \frac{1}{2}}({\mathbf{v}}^{\top }\otimes {\mathbf{v}}^{\top })({\mathbf{\Omega }}^{-1}\otimes {\mathbf{\Omega }}^{-1})]\hfill \\ \hfill & & \times ({\mathbf{I}}_q\otimes {\mathbf{K}}_{n,q}\otimes {\mathbf{I}}_n)({\mathbf{I}}_{q^2}\otimes vec{\mathbf{I}}_n){\mathbf{D}}_q,\hfill \end{array}$$

(6)

where ${\mathbf{K}}_{u,m}$ denotes the $um\times um$ commutation matrix (see e.g., [2]) and ${\mathbf{D}}_m$ denotes the $m^2\times m(m+1)/2$ duplication matrix; cf. [9]. From [9] (Formula (14)), we have $\mathbf{b}\otimes \mathbf{a}=vec\left(\mathbf{a}{\mathbf{b}}^{\top }\right)$, and thus

$${vec}^{\top }{\mathbf{\Omega }}^{-1}-({\mathbf{v}}^{\top }\otimes {\mathbf{v}}^{\top })({\mathbf{\Omega }}^{-1}\otimes {\mathbf{\Omega }}^{-1})={vec}^{\top }[{\mathbf{\Omega }}^{-1}-{\mathbf{\Omega }}^{-1}\mathbf{v}{\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1}].$$

From [8] (Lemma 2.9), we have

$$({\mathbf{I}}_{m^2}\otimes {vec}^{\top }{\mathbf{I}}_u)({\mathbf{I}}_m\otimes {\mathbf{K}}_{u,m}\otimes {\mathbf{I}}_u)vec\mathbf{A}=vec\left[{PTr}_u\mathbf{A}\right],$$

which implies that the normal equations can be written as stated in the theorem. □

From the Fisher-Neyman factorization theorem, it can be seen that the sufficient statistic (3) remains sufficient for estimation of ${\mathbf{B}}_1$, $\mathbf{\Psi }$, and $\mathbf{\Sigma }$ also in the mixed EGCM (4), since the log-likelihood Function (5) can be expressed in terms of $({\mathbf{A}}_1^{\top }\mathbf{Y},{\mathbf{A}}_2^{\top }\mathbf{Y},{\mathbf{Y}}^{\top }\mathbf{Y})$. Indeed, since

$$\begin{array}{ccc}\hfill {\mathbf{\Omega }}^{-1}& =& {\mathbf{\Sigma }}^{-1}\otimes {\mathbf{I}}_n-({\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2^{\top }\otimes {\mathbf{A}}_2){({\mathbf{C}}_2{\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2^{\top }\otimes {\mathbf{A}}_2^{\top }{\mathbf{A}}_2)}^{-1}({\mathbf{C}}_2{\mathbf{\Sigma }}^{-1}\otimes {\mathbf{A}}_2^{\top }),\hfill \end{array}$$

the quadratic form in (5) can be represented as

$$\begin{array}{ccc}\hfill {\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1}\mathbf{v}& =& {\mathbf{v}}^{\top }({\mathbf{\Sigma }}^{-1}\otimes {\mathbf{I}}_n)\mathbf{v}-\mathbf{v}({\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2^{\top }\otimes {\mathbf{A}}_2)\mathbf{W}({\mathbf{C}}_2{\mathbf{\Sigma }}^{-1}\otimes {\mathbf{A}}_2^{\top })\mathbf{v}\hfill \\ & =& tr\left[{\mathbf{V}}^{\top }\mathbf{V}{\mathbf{\Sigma }}^{-1}\right]-{vec}^{\top }\left[{\mathbf{A}}_2^{\top }\mathbf{V}{\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2\right]\mathbf{W}vec\left[{\mathbf{A}}_2^{\top }\mathbf{V}{\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2\right],\hfill \end{array}$$

where $\mathbf{W}={({\mathbf{C}}_2{\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2^{\top }\otimes {\mathbf{A}}_2^{\top }{\mathbf{A}}_2)}^{-1}$ and $\mathbf{V}=\mathbf{Y}-{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1$. Similarly to the case of fixed effects EGCM, ${\mathbf{V}}^{\top }\mathbf{V}$ can be expressed in terms of $({\mathbf{A}}_1^{\top }\mathbf{Y},{\mathbf{Y}}^{\top }\mathbf{Y})$, while ${\mathbf{A}}_2^{\top }\mathbf{V}$ can be expressed in terms of ${\mathbf{A}}_2^{\top }\mathbf{Y}$.

In the next section, we consider a particular form of the mixed effects EGCM, that is, we assume that ${\mathbf{B}}_1$ is a matrix of treatment effects, ${\mathbf{B}}_2$ is a matrix of block effects (the same for all q characteristics), and that the dispersion matrix $\mathbf{\Omega }$ belongs to a particular quadratic subspace.

3. Block Effects Model with Compound Symmetry Matrix $\mathbf{\Sigma }$

Consider an experiment in which t treatments are arranged in b blocks, each of size k. Assume a mixed effects EGCM in which ${\mathbf{B}}_1$ is a $t\times q_1$ matrix of treatment effects and ${\mathbf{B}}_2$ is a $b\times q_2$ matrix of random block effects. The design matrix for treatments, ${\mathbf{A}}_1$, represents the arrangement of treatments on experimental units. It is therefore a 0–1 matrix with $n=bk$ rows and t columns, such that there is exactly one 1 in each row. The design matrix for block effects, ${\mathbf{A}}_2$, represents the arrangement of experimental units in blocks, and hence ${\mathbf{A}}_2={\mathbf{I}}_b\otimes {\mathbf{1}}_k$. Clearly, ${\mathbf{A}}_2^{\top }{\mathbf{A}}_2=k{\mathbf{I}}_b$. Under these assumptions, the inverse of $\mathbf{\Omega }$ can be written as

$$\begin{array}{ccc}\hfill {\mathbf{\Omega }}^{-1}& =& {\mathbf{\Sigma }}^{-1}\otimes {\mathbf{I}}_{bk}-({\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2^{\top }\otimes {\mathbf{A}}_2){({\mathbf{\Psi }}^{-1}\otimes {\mathbf{I}}_b+{\mathbf{C}}_2{\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2^{\top }\otimes {\mathbf{A}}_2^{\top }{\mathbf{A}}_2)}^{-1}({\mathbf{C}}_2{\mathbf{\Sigma }}^{-1}\otimes {\mathbf{A}}_2^{\top })\hfill \\ & =& {\mathbf{\Sigma }}^{-1}\otimes {\mathbf{I}}_{bk}-{\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2^{\top }{({\mathbf{\Psi }}^{-1}+k{\mathbf{C}}_2{\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2^{\top })}^{-1}{\mathbf{C}}_2{\mathbf{\Sigma }}^{-1}\otimes ({\mathbf{I}}_b\otimes {\mathbf{1}}_k{\mathbf{1}}_k^{\top })\hfill \\ & =& {\mathbf{\Sigma }}^{-1}\otimes {\mathbf{I}}_n-\mathbf{\Xi }\otimes ({\mathbf{I}}_b\otimes {\mathbf{1}}_k{\mathbf{1}}_k^{\top }),\hfill \end{array}$$

where $\mathbf{\Xi }={\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2^{\top }{({\mathbf{\Psi }}^{-1}+k{\mathbf{C}}_2{\mathbf{\Sigma }}^{-1}{\mathbf{C}}_2^{\top })}^{-1}{\mathbf{C}}_2{\mathbf{\Sigma }}^{-1}$. Plugging this ${\mathbf{\Omega }}^{-1}$ into the system of equations in Theorem 2, and using the following properties of the partial trace operator:

$${PTr}_m[vec\mathbf{A}·{vec}^{\top }\mathbf{B}]={\mathbf{A}}^{\top }\mathbf{B},{PTr}_m[\mathbf{A}\otimes \mathbf{B}]=tr\left[\mathbf{B}\right]·\mathbf{A}$$

(cf. [8] (Lemmas 2.8 and 2.13)), and noting that

$${\mathbf{\Omega }}^{-1}\mathbf{v}=vec\left[\mathbf{V}{\mathbf{\Sigma }}^{-1}\right]-vec[({\mathbf{I}}_b\otimes {\mathbf{1}}_k{\mathbf{1}}_k^{\top })\mathbf{V}\mathbf{\Xi }],$$

the maximum likelihood equations from Theorem 2 can be written in matrix form as

$$\begin{array}{c}{\mathbf{A}}_1^{\top }\mathbf{V}{\mathbf{\Sigma }}^{-1}{\mathbf{C}}_1^{\top }={\mathbf{A}}_1^{\top }({\mathbf{I}}_b\otimes {\mathbf{1}}_k{\mathbf{1}}_k^{\top })\mathbf{V}\mathbf{\Xi }{\mathbf{C}}_1^{\top },\hfill \\ {\mathbf{C}}_2({\mathbf{\Sigma }}^{-1}-k\mathbf{\Xi }){\mathbf{V}}^{\top }({\mathbf{I}}_b\otimes {\mathbf{1}}_k{\mathbf{1}}_k^{\top })\mathbf{V}({\mathbf{\Sigma }}^{-1}-k\mathbf{\Xi }){\mathbf{C}}_2^{\top }=bk{\mathbf{C}}_2({\mathbf{\Sigma }}^{-1}-k\mathbf{\Xi }){\mathbf{C}}_2^{\top },\hfill \\ ({\mathbf{\Sigma }}^{-1}{\mathbf{V}}^{\top }-\mathbf{\Xi }{\mathbf{V}}^{\top }({\mathbf{I}}_b\otimes {\mathbf{1}}_k{\mathbf{1}}_k^{\top }))(\mathbf{V}{\mathbf{\Sigma }}^{-1}-({\mathbf{I}}_b\otimes {\mathbf{1}}_k{\mathbf{1}}_k^{\top })\mathbf{V}\mathbf{\Xi })=n{\mathbf{\Sigma }}^{-1}-bk\mathbf{\Xi }.\hfill \end{array}$$

Assume now that the same block effect is observable for every characteristic, that is, ${\mathbf{C}}_2={\mathbf{1}}_q^{\top }$. Then ${\mathbf{B}}_2={\mathit{\beta }}_2$, implying that $\mathbf{\Psi }={\psi }^2$. Assume also that the covariance matrix $\mathbf{\Sigma }$ has a compound symmetry structure, i.e., $\mathbf{\Sigma }={\sigma }^2·[(1-\rho ){\mathbf{I}}_q+\rho {\mathbf{1}}_q{\mathbf{1}}_q^{\top }]$, and that ${\mathbf{1}}_q\in \mathcal{C}\left({\mathbf{C}}_1^{\top }\right)$. Then, we can formulate the following theorem.

Theorem 3. 

In a block effects EGCM (4) with ${\mathbf{A}}_2={\mathbf{I}}_b\otimes {\mathbf{1}}_k$, ${\mathbf{C}}_2={\mathbf{1}}_q^{\top }$, ${\mathbf{1}}_q\in \mathcal{C}\left({\mathbf{C}}_1^{\top }\right)$, an unknown positive ${\psi }^2$ and compound symmetry covariance matrix Σ, the maximum likelihood estimators of the unknown parameters are given by

$$\begin{array}{ccc}vec{\mathbf{B}}_1& =& {\left[({\mathbf{C}}_1\otimes {\mathbf{A}}_1^{\top }){\mathbf{\Omega }}^{-1}({\mathbf{C}}_1^{\top }\otimes {\mathbf{A}}_1)\right]}^{-1}({\mathbf{C}}_1\otimes {\mathbf{A}}_1^{\top }){\mathbf{\Omega }}^{-1}\mathrm{vec}\mathbf{Y}\hfill \\ \mathbf{\Omega }& =& \phantom{+}\frac{{\mathbf{v}}^{\top }({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})\mathbf{v}}{b}({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})\hfill \\ & & +\frac{{\mathbf{v}}^{\top }({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{Q}}_{1_k})\mathbf{v}}{b(k-1)}({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{Q}}_{1_k})\hfill \\ & & +\frac{{\mathbf{v}}^{\top }({\mathbf{Q}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{I}}_k)\mathbf{v}}{(q-1)bk}({\mathbf{Q}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{I}}_k)\hfill \end{array}$$

where $\mathbf{v}={\mathbf{Q}}_{(C_1^{\top }\otimes A_1);{\mathbf{\Omega }}^{-1}}vec\mathbf{Y}$.

Proof. 

Observing that $\mathbf{\Sigma }$ can be represented as ${\lambda }_1{\mathbf{Q}}_{1_q}+{\lambda }_2{\mathbf{P}}_{1_q}$, where ${\lambda }_1={\sigma }^2(1-\rho )$ and ${\lambda }_2={\sigma }^2[1+(q-1)\rho ]$, its inverse can be written as ${\mathbf{\Sigma }}^{-1}={\displaystyle \frac{1}{{\lambda }_1}}{\mathbf{Q}}_{1_q}+{\displaystyle \frac{1}{{\lambda }_2}}{\mathbf{P}}_{1_q}$. Thus,

$$\begin{array}{ccc}\hfill \mathbf{\Omega }& =& {\psi }^2·{\mathbf{1}}_q{\mathbf{1}}_q^{\top }\otimes ({\mathbf{I}}_b\otimes {\mathbf{1}}_k{\mathbf{1}}_k^{\top })+({\lambda }_1{\mathbf{Q}}_{1_q}+{\lambda }_2{\mathbf{P}}_{1_q})\otimes {\mathbf{I}}_n\hfill \\ \hfill & =& (qk{\psi }^2+{\lambda }_2)({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})+{\lambda }_2({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{Q}}_{1_k})+{\lambda }_1({\mathbf{Q}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{I}}_k).\hfill \end{array}$$

(7)

Note that the components of $\mathbf{\Omega }$ are orthogonal, and thus

$${\mathbf{\Omega }}^{-1}=\frac{1}{qk{\psi }^2+{\lambda }_2}({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})+\frac{1}{{\lambda }_2}({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{Q}}_{1_k})+\frac{1}{{\lambda }_1}({\mathbf{Q}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{I}}_k).$$

(8)

Therefore, to obtain the normal equation for ${\psi }^2$, it is sufficient to use the above formula for ${\mathbf{\Omega }}^{-1}$ in the second equation of the system in Theorem 2, and replace ${\mathbf{A}}_2$ and ${\mathbf{C}}_2$ with the considered matrices. This yields

$$tr\left[({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})({\mathbf{\Omega }}^{-1}-{\mathbf{\Omega }}^{-1}\mathbf{v}{\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1})\right]=0.$$

Differentiating $\mathbf{\Sigma }$ with respect to ${\lambda }_1$ and ${\lambda }_2$ gives

$${\displaystyle \frac{\partial \mathbf{\Sigma }}{\partial {\lambda }_1}=vec{\mathbf{Q}}_{1_q}\mathrm{and}\frac{\partial \mathbf{\Sigma }}{\partial {\lambda }_2}=vec{\mathbf{P}}_{1_q},}$$

and hence, the third normal equation in the system from Theorem 2 must be multiplied by the above giving

$$\begin{array}{c}{vec}^{\top }\left({PTr}_n[{\mathbf{\Omega }}^{-1}-{\mathbf{\Omega }}^{-1}\mathbf{v}{\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1}]\right)vec{\mathbf{Q}}_{1_q}=0,\hfill \\ {vec}^{\top }\left({PTr}_n[{\mathbf{\Omega }}^{-1}-{\mathbf{\Omega }}^{-1}\mathbf{v}{\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1}]\right)vec{\mathbf{P}}_{1_q}=0,\hfill \end{array}$$

which, using ${vec}^{\top }\mathbf{A}vec\mathbf{B}=tr\left({\mathbf{A}}^{\top }\mathbf{B}\right)$, reduces to

$$\begin{array}{ccc}tr\left[{PTr}_n[{\mathbf{\Omega }}^{-1}-{\mathbf{\Omega }}^{-1}\mathbf{v}{\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1}]{\mathbf{Q}}_{1_q}\right]\hfill & =& 0\hfill \\ tr\left[{PTr}_n[{\mathbf{\Omega }}^{-1}-{\mathbf{\Omega }}^{-1}\mathbf{v}{\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1}]{\mathbf{P}}_{1_q}\right]\hfill & =& 0.\hfill \end{array}$$

Taking into account that ${\mathbf{Q}}_{1_q}={\mathbf{I}}_q-{\mathbf{P}}_{1_q}$ and using the second equality, as well as [8] (Lemma 2.11) and the fact that $tr[PTr[·\left]\right]=tr[·]$, we may rewrite the above as

$$\begin{array}{ccc}tr\left[{\mathbf{\Omega }}^{-1}-{\mathbf{\Omega }}^{-1}\mathbf{v}{\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1}\right]\hfill & =& 0,\hfill \\ tr\left[({\mathbf{\Omega }}^{-1}-{\mathbf{\Omega }}^{-1}\mathbf{v}{\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1})({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_n)\right]\hfill & =& 0.\hfill \end{array}$$

Plugging ${\mathbf{\Omega }}^{-1}$ into the normal equations for ${\psi }^2$, ${\lambda }_1$ and ${\lambda }_2$, and using orthogonality and idempotency of the components of ${\mathbf{\Omega }}^{-1}$, the MLEs of the unknown dispersion parameters under the mixed effects EGCM with additional restrictions can be obtained as

$$\begin{array}{c}{\displaystyle {\psi }^2=\frac{{\mathbf{v}}^{\top }[(k-1)({\mathbf{P}}_q\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_k)-({\mathbf{P}}_q\otimes {\mathbf{I}}_b\otimes {\mathbf{Q}}_k)]\mathbf{v}}{qbk(k-1)}}\hfill \\ {\displaystyle {\lambda }_1=\frac{{\mathbf{v}}^{\top }({\mathbf{Q}}_q\otimes {\mathbf{I}}_{bk})\mathbf{v}}{(q-1)bk}}\hfill \\ {\displaystyle {\lambda }_2=\frac{{\mathbf{v}}^{\top }({\mathbf{P}}_q\otimes {\mathbf{I}}_b\otimes {\mathbf{Q}}_{1_k})\mathbf{v}}{b(k-1)}.}\hfill \end{array}$$

Substituting the above into $\mathbf{\Omega }$ given in (7) we obtain the desired result. □

Note that the positive definiteness of the estimate of $\mathbf{\Omega }$ follows from the positivity of all coefficients of the respective Kronecker products, which are orthogonal to each other and sum to the identity matrix.

It is known (see, e.g., [3,11]) that the sufficient statistics for estimation of ${\mathbf{B}}_1$, ${\mathbf{B}}_2$, and $\mathbf{\Sigma }={\mathbf{\Sigma }}_{CS}$ under the fixed effects EGCM (1), with ${\mathbf{C}}_2={\mathbf{1}}_q^{\top }$ and ${\mathbf{1}}_q\in \mathcal{R}\left({\mathbf{C}}_1^{\top }\right)$, is

$$(tr\left[{\mathbf{Y}}^{\top }\mathbf{Y}\right],{\mathbf{1}}_q^{\top }{\mathbf{Y}}^{\top }\mathbf{Y}{\mathbf{1}}_q,{\mathbf{A}}_1^{\top }\mathbf{Y}{\mathbf{C}}_1^{\top },{\mathbf{A}}_2^{\top }\mathbf{Y}{\mathbf{1}}_q).$$

(9)

It can be shown that this is also a sufficient statistic for the estimation of ${\mathbf{B}}_1$, ${\psi }^2$ and $\mathbf{\Sigma }={\mathbf{\Sigma }}_{CS}$ under the considered mixed effects EGCM. Taking into account (5), it is enough to show that ${\mathbf{v}}^{\top }{\mathbf{\Omega }}^{-1}\mathbf{v}$, with ${\mathbf{\Omega }}^{-1}$ given in (8), is a function of (9). Note that

$$\begin{array}{ccc}\hfill {\mathbf{v}}^{\top }({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})\mathbf{v}& =& tr\left[{\mathbf{V}}^{\top }({\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})\mathbf{V}{\mathbf{P}}_{1_q}\right]=\frac{1}{q}{\mathbf{1}}_q^{\top }{\mathbf{V}}^{\top }({\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})\mathbf{V}{\mathbf{1}}_q\hfill \\ \hfill {\mathbf{v}}^{\top }({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{Q}}_{1_k})\mathbf{v}& =& {\mathbf{v}}^{\top }({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_{bk})\mathbf{v}-{\mathbf{v}}^{\top }({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})\mathbf{v}\hfill \\ & =& tr\left[{\mathbf{V}}^{\top }\mathbf{V}{\mathbf{P}}_{1_q}\right]-tr\left[{\mathbf{V}}^{\top }({\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})\mathbf{V}{\mathbf{P}}_{1_q}\right]\hfill \\ & =& \frac{1}{q}{\mathbf{1}}_q^{\top }{\mathbf{V}}^{\top }\mathbf{V}{\mathbf{1}}_q-\frac{1}{q}{\mathbf{1}}_q^{\top }{\mathbf{V}}^{\top }({\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})\mathbf{V}{\mathbf{1}}_q\hfill \\ & =& \frac{1}{q}{\mathbf{1}}_q^{\top }{\mathbf{V}}^{\top }\mathbf{V}{\mathbf{1}}_q-\frac{1}{kq}{\mathbf{1}}_q^{\top }{\mathbf{V}}^{\top }{\mathbf{A}}_2{\mathbf{A}}_2^{\top }\mathbf{V}{\mathbf{1}}_q,\hfill \end{array}$$

and

$${\mathbf{v}}^{\top }({\mathbf{Q}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{I}}_k)\mathbf{v}={\mathbf{v}}^{\top }\mathbf{v}-{\mathbf{v}}^{\top }({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_{bk})\mathbf{v}=tr\left[{\mathbf{V}}^{\top }\mathbf{V}\right]-\frac{1}{q}{\mathbf{1}}_q^{\top }{\mathbf{V}}^{\top }\mathbf{V}{\mathbf{1}}_q.$$

We therefore need to show that ${\mathbf{1}}_q^{\top }{\mathbf{V}}^{\top }\mathbf{V}{\mathbf{1}}_q$, ${\mathbf{1}}_q^{\top }{\mathbf{V}}^{\top }{\mathbf{A}}_2{\mathbf{A}}_2^{\top }\mathbf{V}{\mathbf{1}}_q$, and $tr\left[{\mathbf{V}}^{\top }\mathbf{V}\right]$ are functions of (9). Consider ${\mathbf{1}}_q^{\top }{\mathbf{V}}^{\top }\mathbf{V}{\mathbf{1}}_q=qtr\left[{\mathbf{V}}^{\top }\mathbf{V}{\mathbf{P}}_{1_q}\right]$. Since ${\mathbf{1}}_q\in \mathcal{C}\left({\mathbf{C}}_1^{\top }\right)$, we have ${\mathbf{P}}_{1_q}={\mathbf{P}}_{C_1^{\top }}{\mathbf{P}}_{1_q}={\mathbf{P}}_{1_q}{\mathbf{P}}_{C_1^{\top }}$. Therefore

$$\begin{array}{ccc}\hfill {\mathbf{1}}_q^{\top }{\mathbf{V}}^{\top }\mathbf{V}{\mathbf{1}}_q& =& {\mathbf{1}}_q^{\top }{\mathbf{Y}}^{\top }\mathbf{Y}{\mathbf{1}}_q-2qtr\left[{\mathbf{C}}_1{\mathbf{P}}_{1_q}{\mathbf{Y}}^{\top }{\mathbf{A}}_1{\mathbf{B}}_1\right]+{\mathbf{1}}_q^{\top }{\mathbf{C}}_1^{\top }{\mathbf{B}}_1^{\top }{\mathbf{A}}_1^{\top }{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1{\mathbf{1}}_q\hfill \\ & =& {\mathbf{1}}_q^{\top }{\mathbf{Y}}^{\top }\mathbf{Y}{\mathbf{1}}_q-2qtr\left[{\mathbf{C}}_1{\mathbf{P}}_{1_q}{\mathbf{P}}_{C_1^{\top }}{\mathbf{Y}}^{\top }{\mathbf{A}}_1{\mathbf{B}}_1\right]+{\mathbf{1}}_q^{\top }{\mathbf{C}}_1^{\top }{\mathbf{B}}_1^{\top }{\mathbf{A}}_1^{\top }{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1{\mathbf{1}}_q\hfill \\ & =& {\mathbf{1}}_q^{\top }{\mathbf{Y}}^{\top }\mathbf{Y}{\mathbf{1}}_q-2{\mathbf{1}}_q^{\top }{\mathbf{C}}_1^{\top }{\left({\mathbf{C}}_1{\mathbf{C}}_1^{\top }\right)}^{-}{\mathbf{C}}_1{\mathbf{Y}}^{\top }{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1{\mathbf{1}}_q+{\mathbf{1}}_q^{\top }{\mathbf{C}}_1^{\top }{\mathbf{B}}_1^{\top }{\mathbf{A}}_1^{\top }{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1{\mathbf{1}}_q\hfill \end{array}$$

is a function of ${\mathbf{1}}_q^{\top }{\mathbf{Y}}^{\top }\mathbf{Y}{\mathbf{1}}_q$ and ${\mathbf{A}}_1^{\top }\mathbf{Y}{\mathbf{C}}_1^{\top }$.

It can be easily seen that

$$\begin{array}{ccc}\hfill {\mathbf{1}}_q^{\top }{\mathbf{V}}^{\top }{\mathbf{A}}_2{\mathbf{A}}_2^{\top }\mathbf{V}{\mathbf{1}}_q& =& {\mathbf{1}}_q^{\top }{\mathbf{Y}}^{\top }{\mathbf{A}}_2{\mathbf{A}}_2^{\top }\mathbf{Y}{\mathbf{1}}_q-2{\mathbf{1}}_q^{\top }{\mathbf{Y}}^{\top }{\mathbf{A}}_2{\mathbf{A}}_2^{\top }{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1{\mathbf{1}}_q\hfill \\ & & +{\mathbf{1}}_q^{\top }{\mathbf{A}}_1^{\top }{\mathbf{B}}_1^{\top }{\mathbf{C}}_1\top {\mathbf{A}}_2{\mathbf{A}}_2^{\top }{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1{\mathbf{1}}_q\hfill \end{array}$$

is a function of ${\mathbf{A}}_2^{\top }\mathbf{Y}{\mathbf{1}}_q$, and that

$$tr\left[\mathbf{V}{\mathbf{V}}^{\top }\right]=tr\left[{\mathbf{Y}}^{\top }\mathbf{Y}\right]-2tr\left[{\mathbf{C}}_1{\mathbf{Y}}^{\top }{\mathbf{A}}_1{\mathbf{B}}_1\right]+tr\left[{\mathbf{C}}_1^{\top }{\mathbf{B}}_1^{\top }{\mathbf{A}}_1^{\top }{\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1\right]$$

is a function of $tr\left[{\mathbf{Y}}^{\top }\mathbf{Y}\right]$ and ${\mathbf{A}}_1^{\top }\mathbf{Y}{\mathbf{C}}_1^{\top }$, and thus sufficiency follows.

Finally, let us take a look at the design matrix for treatments, ${\mathbf{A}}_1$, since in block experiments it contains information about the arrangement of treatments on experimental units. The most common block designs are balanced incomplete block designs (BIBDs), that is, binary block designs with $k\le t$, such that each treatment appears in the same number of blocks, say $r_1$, and each pair of distinct treatments appears in the same number of blocks, say $r_2$. Since the design is binary, all treatments within a single block must be distinct; cf. [12]. An immediate consequence is that ${\mathbf{A}}_1^{\top }{\mathbf{A}}_1=(r_1-r_2){\mathbf{I}}_t+r_2{\mathbf{1}}_t{\mathbf{1}}_t^{\top }$. Moreover, in the particular case when $k=t$, the BIBD becomes complete, and ${\mathbf{A}}_1$ can be represented as a partitioned matrix consisting of b permutation matrices, ${\mathbf{\Pi }}_i$, $i=1,\dots ,b$, that is,

$${\mathbf{A}}_1={({\mathbf{\Pi }}_1,{\mathbf{\Pi }}_2,\dots ,{\mathbf{\Pi }}_b)}^{\top }.$$

Obviously, ${\mathbf{A}}_1^{\top }{\mathbf{A}}_1=b{\mathbf{I}}_t$ and ${\mathbf{\Pi }}_i{\mathbf{1}}_t={\mathbf{\Pi }}_i^{\top }{\mathbf{1}}_t={\mathbf{1}}_t$. Since ${\mathbf{P}}_{C_1^{\top }\otimes A_1}={\mathbf{P}}_{C_1^{\top }}\otimes {\mathbf{P}}_{A_1}$ and ${\mathbf{1}}_q\in \mathcal{C}\left({\mathbf{C}}_1^{\top }\right)$, for the dispersion matrix with compound symmetry structure of $\mathbf{\Sigma }$ we have

$${\mathbf{P}}_{C_1^{\top }\otimes A_1}\mathbf{\Omega }=\mathbf{\Omega }{\mathbf{P}}_{C_1^{\top }\otimes A_1},$$

(10)

which means that the orthogonal projector onto the expectation space commutes with the dispersion matrix. Moreover, in such a case, the best linear unbiased estimator (BLUE) of ${\mathbf{B}}_1$ exists and coincides with the ordinary least squares estimator (OLSE); cf. [13]. Indeed, the normal equation obtained from the derivative with respect to ${\mathit{\beta }}_1$ given in (6), that is

$$({\mathbf{C}}_1\otimes {\mathbf{A}}_1^{\top }){\mathbf{\Omega }}^{-1}\mathbf{v}=\mathbf{0}$$

when multiplied on the left by $({\mathbf{C}}_1^{\top }\otimes {\mathbf{A}}_1){\left[({\mathbf{C}}_1\otimes {\mathbf{A}}_1^{\top })({\mathbf{C}}_1^{\top }\otimes {\mathbf{A}}_1)\right]}^{-1}$, this can be represented as

$${\mathbf{P}}_{C_1^{\top }\otimes A_1}{\mathbf{\Omega }}^{-1}\mathbf{v}=\mathbf{0}.$$

From the commutativity relation (10), we obtain an explicit solution for the maximum likelihood estimator of ${\mathbf{B}}_1$ in the form

$${\mathbf{A}}_1{\widehat{\mathbf{B}}}_1{\mathbf{C}}_1={\mathbf{P}}_{A_1}\mathbf{Y}{\mathbf{P}}_{C_1^{\top }},$$

which is also the BLUE of $\left({\mathbf{A}}_1{\mathbf{B}}_1{\mathbf{C}}_1\right)$.

Moreover, since $\mathbf{\Omega }$ belongs to a quadratic subspace, according to [14], the MLEs of the parameters of model (4) have an explicit form, and the estimator of the dispersion matrix can be represented as a projection of $\mathbf{S}={\mathbf{Q}}_{C_1^{\top }\otimes A_1}\mathbf{y}{\mathbf{y}}^{\top }{\mathbf{Q}}_{C_1^{\top }\otimes A_1}$ onto the space of compound symmetry matrices, that is,

$$\begin{array}{ccc}\hfill \widehat{\mathbf{\Omega }}& =& \phantom{+}{\displaystyle \frac{1}{tr({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})}}tr\left[\mathbf{S}({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})\right]·({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})\hfill \\ & & +{\displaystyle \frac{1}{tr({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{Q}}_{1_k})}}tr\left[\mathbf{S}({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{Q_k})\right]·({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{Q}}_{1_k})\hfill \\ & & +{\displaystyle \frac{1}{tr({\mathbf{Q}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{I}}_k)}}tr\left[\mathbf{S}({\mathbf{Q}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{I}}_k)\right]·({\mathbf{Q}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{I}}_k).\hfill \end{array}$$

Using the commutativity relation (10) and the fact that ${\mathbf{1}}_q\in \mathcal{C}\left({\mathbf{C}}_1^{\top }\right)$, we finally obtain the MLEs of ${\mathbf{B}}_1$ and $\mathbf{\Omega }$ for balanced complete block designs as

$$\begin{array}{ccc}{\mathbf{A}}_1{\widehat{\mathbf{B}}}_1{\mathbf{C}}_1& =& {\mathbf{P}}_{A_1}\mathbf{Y}{\mathbf{P}}_{C_1^{\top }}\hfill \\ \widehat{\mathbf{\Omega }}& =& \phantom{+}{\displaystyle \frac{1}{b}}tr\left[{\mathbf{P}}_{1_q}{\mathbf{Y}}^{\top }({\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k}){\mathbf{Q}}_{A_1}\mathbf{Y}\right]·({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{P}}_{1_k})\hfill \\ & & +{\displaystyle \frac{1}{b(k-1)}}tr\left[{\mathbf{P}}_{1_q}{\mathbf{Y}}^{\top }({\mathbf{I}}_b\otimes {\mathbf{Q}}_{1_k}){\mathbf{Q}}_{A_1}\mathbf{Y}\right]·({\mathbf{P}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{Q}}_{1_k})\hfill \\ & & +{\displaystyle \frac{1}{(q-1)bk}}tr[{\mathbf{Q}}_{1_q}{\mathbf{Y}}^{\top }\mathbf{Y}-{\mathbf{Q}}_{1_q}{\mathbf{Y}}^{\top }{\mathbf{P}}_{A_1}\mathbf{Y}{\mathbf{P}}_{C_1^{\top }}]·({\mathbf{Q}}_{1_q}\otimes {\mathbf{I}}_b\otimes {\mathbf{I}}_k).\hfill \end{array}$$

(11)

Note that (11) can be also obtained by substituting $\widehat{\mathbf{v}}=vec[\mathbf{Y}-{\mathbf{P}}_{A_1}\mathbf{Y}{\mathbf{P}}_{C_1^{\top }}]$ instead of $\mathbf{v}$ in Theorem 3.

In many experiments, it is common to replicate observations. In such cases, complete block designs can be generalized to block designs, in which each treatment appears more than once. For example, if each treatment is replicated r times in every block (of size $k=rt$), and if the observations are arranged consecutively with respect to replications, then ${\mathbf{A}}_1={({\mathbf{\Pi }}_{11},{\mathbf{\Pi }}_{12},\dots ,{\mathbf{\Pi }}_{1r},\dots ,{\mathbf{\Pi }}_{b1},{\mathbf{\Pi }}_{b2},\dots ,{\mathbf{\Pi }}_{br})}^{\top }$. This implies that ${\mathbf{A}}_1^{\top }{\mathbf{A}}_1=br{\mathbf{I}}_t$ and ${\mathbf{\Pi }}_{ij}{\mathbf{1}}_t={\mathbf{\Pi }}_{ij}^{\top }{\mathbf{1}}_t={\mathbf{1}}_t$, $i\in \{1,\dots ,b\}$, $j\in \{1,\dots ,r\}$. Note, however, that the commutativity condition (10) still holds, and the MLEs of ${\mathbf{B}}_1$ and $\mathbf{\Sigma }$ can be computed via (11) with ${\mathbf{A}}_1$ given above.

4. Applications

In this section, we illustrate the results of Section 3 by analyzing two subsets of a metabolomic dataset containing measurements of the concentrations of 104 secondary metabolites in 9 barley varieties (C—CamB1, G—Georgia, H—Harmal, L—Lubuski, M—Maresi, Md—MDingo, Mx—Morex, S—Sebastian, St—Stratus) under 4 different environmental conditions related to water shortage: two droughts (I and II), a combined drought (I + II), and a control condition. Each variety was observed at 7 time points (from T2 to T8), in 4 biological replicates (plants). The data were collected at the Institute of Plant Genetics, Polish Academy of Sciences, in Poznań, in a pilot study for a larger systems biology project aimed at investigating the effects of water shortage on the concentrations of secondary metabolites in the leaves of barley plants; cf. [15].

In the examples below, we focus on the fixed effects of variety, treating the drought effects as random nuisance parameters, since the three droughts and the control condition represent only a subset of possible environmental conditions. Two models that differ in the response structure but share the same random component are considered.

Example 1. 

In this example, we analyze the concentration of the first metabolite in the dataset under three environmental conditions (II, I + II, and a control), at three time points ($q=3$) at the end of the experiment (T6, T7, T8). All four biological observations ($r=4$) of all nine varieties ($t=9$) are taken into account. We are interested in the linear trend in metabolite concentration over time for each variety, taking into account that the environmental conditions may additionally influence the results. Considering these environmental conditions as random blocks ($b=3$) of size $k=rt$, and assuming that the drought condition affects the response equally at each time point, we employ the random effects EGCM (4) with a $bk\times q$ matrix of observations, $\mathbf{Y}$. Specifically, the rows of the observation matrix $\mathbf{Y}$ are arranged first by block (drought condition) and, within each block, by replicate, with varieties nested within replicates. Thus, the observations corresponding to the first drought condition appear first, followed by those from the second and third conditions. Within each block, all nine varieties are recorded sequentially for the first replicate, then for the second, and so on. This ordering implies that

$${\mathbf{A}}_1={\mathbf{1}}_{rb}\otimes {\mathbf{I}}_{t}and{\mathbf{A}}_2={\mathbf{I}}_b\otimes {\mathbf{1}}_k.$$

Considering the linear trend in metabolite concentration over time, we assume that

$${\mathbf{C}}_1=\left(\begin{array}{ccc}1& 1& 1\\ 3& 6& 10\end{array}\right).$$

The second row represents the numerical coding of time: third, sixth and 10th day of the second part of the experiment. This coding implies that the coefficient associated with the second row of ${\mathbf{C}}_1$ in ${\mathbf{B}}_1$ represents the rate of change in metabolite concentration per time unit. The assumption of an equal drought effect across time points leads to

$${\mathbf{C}}_2={\mathbf{1}}_q^{\top },$$

reducing the covariance matrix of the random drought effects to a scalar, ${\psi }^2$. Finally, since all measurements were taken under comparable experimental conditions and within a relatively short time window, it is reasonable to assume that the correlations between time points are approximately equal. Therefore, we assume that metabolite concentrations at a single time point have equal variances and identical covariances at distinct time points, resulting in a compound symmetry structure of the $3\times 3$ matrix Σ.

Using Theorem 3 we obtain

$$\widehat{\mathbf{B}}=\left(\begin{array}{cc}\hfill 21.0097& \hfill 0.0609\\ \hfill 23.4933& \hfill -0.0042\\ \hfill 23.4382& \hfill -0.0688\\ \hfill 23.1798& \hfill -0.0659\\ \hfill 21.7026& \hfill 0.0243\\ \hfill 22.1970& \hfill 0.0379\\ \hfill 22.3129& \hfill 0.0945\\ \hfill 23.4419& \hfill -0.0590\\ \hfill 23.5034& \hfill -0.0067\end{array}\right)$$

and

$$\widehat{\mathbf{\Omega }}=3.4871({\mathbf{P}}_{1_3}\otimes {\mathbf{I}}_3\otimes {\mathbf{P}}_{1_{36}})+1.2519({\mathbf{P}}_{1_3}\otimes {\mathbf{I}}_3\otimes {\mathbf{Q}}_{1_{36}})+1.2570({\mathbf{Q}}_{1_3}\otimes {\mathbf{I}}_{108})$$

with the estimate of the variance of block effects given by ${\widehat{\psi }}^2=0.0207$ and

$$\widehat{\mathbf{\Sigma }}=\left(\begin{array}{ccc}\hfill 1.2553& \hfill -0.0017& \hfill -0.0017\\ \hfill -0.0017& \hfill 1.2553& \hfill -0.0017\\ \hfill -0.0017& \hfill -0.0017& \hfill 1.2553\end{array}\right),$$

implying ${\widehat{\sigma }}^2=1.2553$ and $\widehat{\rho }=-0.0014$.

The estimated fixed effects matrix ${\widehat{\mathbf{B}}}_1$ indicates that the mean concentration of the first metabolite varies moderately across varieties, with estimated intercepts ranging from 21.0097 to 23.5034. The estimated slopes, corresponding to the linear trend over time, are close to zero for most varieties and vary in sign, suggesting the absence of a consistent overall increase or decrease in metabolite concentration during the experiment.

The estimated variance components are ${\widehat{\psi }}^2=0.0207$ for the random drought effect and ${\widehat{\sigma }}^2=1.2553$ for the residual variance. The small value of ${\widehat{\psi }}^2$ indicates that the environmental (drought) conditions contribute little to the total variability in metabolite concentration, confirming that differences among varieties dominate over drought effects.

The estimated correlation parameter $\widehat{\rho }=-0.0014$ suggests that correlations between time points are negligible, which supports the assumption of weak temporal dependence within each variety.

Performed analysis suggests that the concentration of the metabolite remained stable over the experimental period, and that the drought treatments did not have a substantial impact compared with varietal differences.

Recall that ${\widehat{\mathbf{B}}}_1$ is the best linear unbiased estimator of ${\mathbf{B}}_1$, while the MLEs of ${\psi }^2$, ${\sigma }^2$, and $\varrho$ are biased. To assess the accuracy of the estimates for different sample sizes, we conduct small simulation studies using parameters near the estimated ones. Specifically, we generate 10,000 data matrices from the matrix normal distribution with expectation

$${\mathbf{A}}_1{\widehat{\mathbf{B}}}_1{\mathbf{C}}_1$$

and dispersion matrix

$${\widehat{\psi }}^2{\mathbf{C}}_2^{\top }{\mathbf{C}}_2\otimes {\mathbf{A}}_2{\mathbf{A}}_2^{\top }+\widehat{\mathbf{\Sigma }}\otimes {\mathbf{I}}_n,$$

where

$$\widehat{\mathbf{\Sigma }}={\widehat{\sigma }}^2\left[(1-\widehat{\rho }){\mathbf{I}}_q+\widehat{\rho }{\mathbf{1}}_q{\mathbf{1}}_q^{\top }\right],$$

and we compute the MLEs of ${\mathbf{B}}_1$, ${\psi }^2$, ${\sigma }^2$, and $\rho$ for each simulated dataset. Note that in such a demanding setup, the only parameter that can be varied is the number of replicates, r. In

Table 1. Empirical bias of the estimated parameters for $r\in \{1,2,4,8\}$ obtained from 10,000 Monte Carlo runs.

	$r=1$	$r=2$
${\psi }^2$	−0.0062	−0.0104
${\sigma }^2$	−0.2791	−0.1347
$\rho$	−0.0728	−0.0282
${\mathbf{B}}_1$	$\left(\begin{array}{cc}\hfill 0.0223& \hfill -0.0023\\ \hfill -0.0040& \hfill 0.0005\\ \hfill -0.0126& \hfill 0.0016\\ \hfill -0.0101& \hfill 0.0011\\ \hfill -0.0163& \hfill 0.0019\\ \hfill -0.0231& \hfill 0.0024\\ \hfill -0.0012& \hfill -0.0006\\ \hfill -0.0030& \hfill -0.0007\\ \hfill -0.0040& \hfill 0.0012\end{array}\right)$	$\left(\begin{array}{cc}\hfill 0.0022& \hfill -0.0001\\ \hfill 0.0028& \hfill -0.0009\\ \hfill -0.0032& \hfill -0.0003\\ \hfill 0.0120& \hfill -0.0015\\ \hfill 0.0050& \hfill -0.0005\\ \hfill -0.0114& \hfill 0.0014\\ \hfill -0.0007& \hfill -0.0004\\ \hfill -0.0006& \hfill -0.0004\\ \hfill 0.0005& \hfill -0.0003\end{array}\right)$
	$r=4$	$r=8$
${\psi }^2$	−0.0095	−0.0086
${\sigma }^2$	−0.0667	−0.0337
$\rho$	−0.0133	−0.0064
${\mathbf{B}}_1$	$\left(\begin{array}{cc}\hfill -0.0013& \hfill 0.0003\\ \hfill -0.0053& \hfill 0.0010\\ \hfill -0.0049& \hfill 0.0004\\ \hfill 0.0008& \hfill -0.0004\\ \hfill -0.0072& \hfill 0.0008\\ \hfill -0.0022& \hfill 0.0008\\ \hfill 0.0000& \hfill -0.0003\\ \hfill -0.0062& \hfill 0.0005\\ \hfill 0.0022& \hfill 0.0001\end{array}\right)$	$\left(\begin{array}{cc}\hfill 0.0000& \hfill 0.0005\\ \hfill -0.0033& \hfill 0.0003\\ \hfill -0.0038& \hfill 0.0005\\ \hfill 0.0034& \hfill -0.0007\\ \hfill -0.0040& \hfill 0.0003\\ \hfill -0.0034& \hfill 0.0004\\ \hfill 0.0007& \hfill -0.0001\\ \hfill -0.0091& \hfill 0.0010\\ \hfill 0.0038& \hfill -0.0004\end{array}\right)$

, we demonstrate the empirical bias of the estimated parameters obtained from 10,000 Monte Carlo runs for four different values of r.

Results presented in Table 1 indicate that the MLEs of the variance components are biased, particularly for small sample sizes. The bias of ${\widehat{\sigma }}^2$ is notably larger than that of ${\widehat{\psi }}^2$ and $\widehat{\rho }$, but it decreases substantially as the number of replicates r increases. The estimator of the correlation parameter $\widehat{\rho }$ shows only a small negative bias even for $r=1$, which becomes negligible for larger r. The estimator ${\widehat{\mathbf{B}}}_1$ is theoretically unbiased, and the small deviations observed in the simulation results can be attributed to sampling variability inherent in the Monte Carlo procedure. Overall, the results confirm that the accuracy of the estimators improves with increasing sample size, as expected from the asymptotic properties of maximum likelihood estimators.

Example 2. 

In this example, we analyze the concentrations of all 104 metabolites measured at the end of the experiment (T8) under the same three environmental conditions (b = 3) as in previous example. The dataset includes all nine barley varieties (t = 9), each represented by four biological replicates (r = 4). The aim is to investigate the effect of variety on metabolite concentrations while accounting for the variability introduced by the environmental conditions. Treating the drought conditions as random blocks of size $k=rt$, and assuming that the drought effect is constant across all metabolites, we employ the random effects EGCM (4) with a $bk\times q$ matrix of observations, $\mathbf{Y}$, where q = 104.

Similarly as in Example 1, the rows of the observation matrix $\mathbf{Y}$ are arranged first by block (drought condition) and, within each block, by replicate, with varieties nested within replicates. Consequently, the treatment and block design matrices are the same as before, that is

$${\mathbf{A}}_1={\mathbf{1}}_{rb}\otimes {\mathbf{I}}_{t}and{\mathbf{A}}_2={\mathbf{I}}_b\otimes {\mathbf{1}}_k.$$

Since the analysis concerns metabolite concentrations observed at a single time point, the corresponding matrix is

$${\mathbf{C}}_1={\mathbf{I}}_q,$$

with $q=104$.

Assuming that the drought effect is identical for all metabolites leads to

$${\mathbf{C}}_2={\mathbf{1}}_q^{\top },$$

which reduces the covariance matrix of random drought effects to a scalar, ${\psi }^2$. Finally, we assume that all metabolites have equal variances and identical pairwise correlations, resulting in a compound symmetry structure of the $104\times 104$ matrix Σ. This assumption is further supported by the heatmap of pairwise correlations between metabolites (Figure 1). The plot reveals a relatively uniform correlation structure, with no distinct clusters or strong block patterns, suggesting that the correlations between metabolites are approximately constant across pairs.

Using Theorem 3, we obtain the estimate ${\widehat{\mathbf{B}}}_1$ of size $9\times 104$, which is therefore not presented in detail. However,

Table 2. Estimated mean concentrations of selected metabolites across nine barley varieties. Left: metabolites with minimal variation between varieties. Right: metabolites with maximal variation between varieties.

variety	vtrait[9]	vtrait[84]	vtrait[87]	vtrait[25]	vtrait[30]	vtrait[31]
1	27.8508	22.8692	24.3817	24.8125	28.9858	26.8550
2	22.4475	23.2183	24.6992	21.7183	28.4525	29.4067
3	23.0925	23.5933	24.7342	22.5092	28.6792	28.9925
4	22.5083	23.1550	24.5750	21.6408	28.2775	28.8625
5	23.2758	23.0300	24.7800	24.9775	28.4075	27.7183
6	22.6392	23.1475	24.7942	21.9442	28.3717	28.9525
7	22.3358	22.7808	24.4058	21.8908	28.5150	29.3667
8	22.5292	23.1825	24.9267	21.9033	28.7233	29.3100
9	22.8033	22.6975	24.4250	21.6517	28.6667	29.8375

shows the estimated mean concentrations of three metabolites with minimal variation and three metabolites with maximal variation across the nine barley varieties. Metabolites vtrait[87], vtrait[84], and vtrait[9] show very small differences in their estimated means, indicating that their levels are largely unaffected by genotype. Such metabolites are likely stable markers and may not contribute to varietal differentiation. In contrast, metabolites vtrait[25], vtrait[31], and vtrait[30] exhibit substantial differences in estimated mean concentrations between varieties, suggesting a strong genotypic effect. These metabolites are likely important in distinguishing the metabolic profiles of barley varieties and could be prioritized in further studies of varietal traits.

The estimate of Ω is given by

$$\widehat{\mathbf{\Omega }}=20.4000({\mathbf{P}}_{1_{104}}\otimes {\mathbf{I}}_3\otimes {\mathbf{P}}_{1_{36}})+16.0083({\mathbf{P}}_{1_{104}}\otimes {\mathbf{I}}_3\otimes {\mathbf{Q}}_{1_{36}})+0.7131({\mathbf{Q}}_{1_{104}}\otimes {\mathbf{I}}_{108}),$$

which is positive definite. The small value of ${\widehat{\psi }}^2=0.0012$ suggests that the environmental (drought) conditions contribute only marginally to the total variability in metabolite concentration. This indicates that differences among barley varieties dominate over drought-related effects, and the random block component has a negligible impact on the model.

The estimate of the $104\times 104$ compound symmetry covariance matrix of metabolite concentrations has diagonal elements equal to $0.8602$ and off-diagonal elements equal to $0.1471$. Consequently, ${\widehat{\sigma }}^2=0.8602$ and $\widehat{\rho }=0.1710$, which indicates a relatively weak positive correlation among metabolites measured under the same environmental conditions.

The two examples presented above illustrate the flexibility of the EGCM in handling data with different response structures while maintaining a common specification of random effects. In the first example, the model captured the linear trend of metabolite concentration over time for each barley variety, under the assumption of a random block effect representing environmental conditions. In the second example, the same random component structure was applied to a large-dimensional response comprising 104 metabolites measured at the end of the experiment. In both cases, the assumption of compound symmetry for the covariance matrix $\mathbf{\Sigma }$ provided a reasonable and interpretable representation of the dependence structure between observations. However, the near-zero estimate of ${\psi }^2$ in the second example suggests that the random block effects may not play a substantial role in explaining the observed variability. Overall, these analyses demonstrate that the EGCM framework offers a unified approach for modeling complex experimental designs, accommodating both longitudinal and multivariate response structures within a coherent covariance formulation.

5. Conclusions

In this paper, maximum likelihood estimation in the mixed effects EGCM was investigated under the assumption of multivariate normality. The likelihood equations were determined for a general dispersion matrix as well as for models with specific assumptions on the nuisance parameters (block effects). It was demonstrated that, for a compound symmetry structure of the dispersion matrix, these equations simplify considerably. Moreover, for experiments designed in balanced complete blocks, the MLEs of the unknown parameters can be obtained in closed form.

In addition, we demonstrated an analogy between fixed and mixed effects EGCM with respect to sufficiency. This result implies that the original datasets can be reduced to sufficient statistics, which enables more efficient data analysis, as only matrices of substantially lower dimensions are involved in the estimation process. Furthermore, an additional advantage of using sufficient statistics lies in the facilitation of data storage and transfer.

The practical applicability of the proposed framework was illustrated through two examples involving metabolomic data from a barley drought experiment. These analyses confirmed that the EGCM approach can effectively accommodate both longitudinal and large-dimensional multivariate response structures within a unified modeling framework. The results also highlighted that, in some cases, the random block component may have negligible influence, suggesting that model simplification could be justified without loss of interpretability.

Overall, the EGCM framework provides a flexible and computationally tractable tool for modeling complex experimental designs, offering both theoretical clarity and practical efficiency.